Two-element f-θ lens used for micro-electro mechanical system (MEMS) laser scanning unit

ABSTRACT

A two-element f-θ lens used for a micro-electro mechanical system (MEMS) laser scanning unit includes a first lens and a second lens, the first lens is a positive power meniscus lens of which concave surface is disposed on a side of a MEMS mirror, the second lens is a negative power meniscus lens of which convex surface is disposed on the side of the MEMS mirror, at least one optical surface is an Aspherical surface in both main scanning direction and sub scanning direction, and satisfies special optical conditions. The two-element f-θ lens corrects the nonlinear relationship between scanned angle and time into the linear relationship between the image spot distances and time. Meanwhile, the two-element f-θ lens focuses the scan light to the target in the main scanning and sun scanning directions, such that the purpose of the scanning linearity effect and the high resolution scanning can be achieved.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a two-element fθ lens used for amicro-electro mechanical system (MEMS) laser scanning (LSU), and moreparticularly to a two-element fθ lens using an angular change varyingwith time in a sinusoidal relation for correcting a MEMS reflectingmirror having a simple harmonic movement to achieve the scanninglinearity effect by the laser scanning unit.

2. Description of the Related Art

At present, a laser scanning unit (LSU) used by a laser beam printer(LBP) controls a laser beam scanning by a high-speed rotating polygonmirror as disclosed in U.S. Pat. Nos. 7,079,171, 6,377,293 and 6,295,116or Taiwan (R.O.C.) Pat No.I198966, and principles of those inventionsare described as the following: a semiconductor laser emits a laser beamthrough a collimator and an aperture to form parallel beams. After theparallel beams pass through a cylindrical lens, the beams are focused atthe width of the X-axis in the sub scanning direction and along adirection parallel to the Y-axis of the main scanning direction to forma line image and projected onto a high-speed rotating polygon mirror.The polygon mirror includes a plurality of continuous reflecting mirrorsdisposed substantially at or proximate to the focusing position of theline image. The polygon mirror is provided for controlling the directionof projecting the laser beam, so that when a plurality of continuousreflecting mirrors are rotated at high speed, the laser beam projectedonto a reflecting mirror can be extended in a direction parallel to themain scanning direction (Y-axis) at the same angular velocity anddeviated from and reflected onto a fθ linear scanning lens. The fθlinear scanning lens is installed next to the polygon mirror and may beeither a single-element lens structure (or a single scanning lens) or atwo-element lens structure. The function of this fθ linear scanning lensis to focus a laser beam reflected by the reflecting mirror of thepolygon mirror and projected onto the fθ lens into an oval spot that isprojected onto a photoreceptor (or a photoreceptor drum, which is animage surface) to achieve the requirement of the scanning linearity.However, the traditional laser scanning unit LSU still has the followingdrawbacks in its practical use.

(1) The manufacture of the rotating polygon mirror incurs a high levelof difficulty and a high cost, and thus increasing the manufacturingcost of the LSU.

(2) The polygon mirror requires a function of a high-speed rotation(such as 40000 rpm) and a high precision, and thus a cylindrical lens isrequired and installed to the traditional LSU since the width of thegeneral polygon mirror along the Y-axis of the reflecting surface of themirror is very thin, so that the laser beam pass through the cylindricallens can be focused and concentrated into a line (or a spot on theY-axis) and projected onto the reflecting mirror of the polygon mirror.Such arrangement increases the number of components and also complicatesthe assembling operation procedure.

(3) The traditional polygon mirror requires a high-speed rotation (suchas 40000 rpm), and thus noise level is raised. Furthermore, the polygonmirror takes a longer time to be accelerated from a starting speed to anoperating speed, and thus increasing the booting time of the laserscanning.

(4) In the fabrication of the traditional LSU, the central axis of alaser beam projected onto the reflecting mirror of the polygon mirror isnot aligned precisely with the central rotating axis of the polygonmirror, so that it is necessary to take the off axis deviation of thepolygon mirror into consideration for the design of the fθ lens, andthus increasing the difficulty of design and manufacturing the fθ lens.

In recent years, an oscillatory MEMS reflecting mirror is introduced toovercome the shortcomings of the traditional LSU assembly and replacethe laser beam scanning controlled by the traditional polygon mirror.The surface of a torsion oscillator of the MEMS reflecting mirrorcomprises a reflecting layer, and the reflecting layer is oscillated forreflecting the light and further for the scanning. In the future, sucharrangement will be applied in a laser scanning unit (LSU) of an imagingsystem, a scanner or a laser printer, and its scanning efficiency ishigher than the traditional rotating polygon mirror. As disclosed in theU.S. Pat. Nos. 6,844,951 and 6,956,597, at least one driving signal isgenerated, and its driving frequency approaches the resonant frequencyof a plurality of MEMS reflecting mirrors, and the driving signal drivesthe MEMS reflecting mirror to produce a scanning path. In U.S. Pat. Nos.7,064,876, 7,184,187, 7,190,499, 2006/0033021, 2007/0008401 and2006/0279826 or Taiwan (R.O.C.) Pat No. TW 253133, or Japan Pat. No.2006-201350, a MEMS reflecting mirror installed between a collimator anda fθ lens of a LSU module replaces the traditional rotating polygonmirror for controlling the projecting direction of a laser beam. TheMEMS reflecting mirror features the advantages of small components, fastrotation, and low manufacturing cost. However, after the MEMS reflectingmirror is driven by the received voltage for a simple harmonic with asinusoidal relation of time and angular speed, and a laser beamprojected on the MEMS reflecting mirror is reflected with a relation ofreflecting angle θ(t) and time as followsθ(t)=θ_(s)·sin(2π·f·t)  (1)

wherein, f is the scanning frequency of the MEMS reflecting mirror andθ_(s) is the maximum scanning angle at a single side (symmetrical withthe optical Z axis) after the laser beam passes through the MEMSreflecting mirror.

In the same time interval Δt, the corresponding variation of thereflecting angle is not the same but decreasing, and thus constituting asinusoidal relation with time. In other words, the variation of thereflecting angle in the same time interval Δt isΔθ(t)=θ_(s)·(sin(2π·f·t₁)−sin(2π·f·t₂)), which constitutes a non-linearrelation with time. If the reflected light is projected onto the targetfrom a different angle, the distance from the spot will be different inthe same time interval due to the different angle.

Since the angle of the MEMS reflecting mirror situated at a peak and avalley of a sine wave varies with time, and the rotating movements if atraditional polygon mirror are at a constant angular speed, if atraditional fθ lens is installed on a laser scanning unit (LSU) of theMEMS reflecting mirror, the angle of the MEMS reflecting mirror producedby the sinusoidal relation varied with time cannot be corrected, so thatthe speed of laser beam projected on an image side will not anon-uniform speed scanning, and the image on the image side will bedeviated. Therefore, the laser scanning unit or the MEMS laser scanningunit (MEMS LSU) composed of MEMS reflecting mirrors has a characteristicthat after the laser beam is scanned by the MEMS reflecting mirror, scanlights at different angles are formed in the same time. Thus, finding away of developing a fθ lens (some prior art named as f−sin θ lens) forthe MEMS laser scanning unit to correct the scan lights, such that acorrect image will be projected onto the light, examples as, U.S. Pat.No. 7,184,187 provided a polynomial surface for fθ lens to amend theangular velocity variation in the main-scanning direction only. However,the laser light beam is essential an oval-like shape of the crosssection that corrects the scan lights in the main-scanning directiononly may not be achieve the accuracy requirement. Since, a fθ lens withmain-scanning direction correcting as well as sub-scanning directioncorrecting demands immediate attentions and feasible solutions.

SUMMARY OF THE INVENTION

The primary objective of the present invention is to overcome theshortcomings of the prior art by providing a two-element lens used for amicro-electro mechanical system (MEMS) laser scanning unit, whichcomprises a first lens in positive power meniscus shape having a concavesurface on a side of a MEMS mirror, and a second lens in a negativepower meniscus lens having a convex surface on the side of the MEMSmirror, counted from the MEMS reflecting mirror, for projecting a scanlight reflected by the MEMS reflecting mirror onto the correct image ofa target to achieve a scanning linearity effect required by the laserscanning unit.

Another objective of the present invention is to provide a two-elementfθ lens used for a micro-electro mechanical system (MEMS) laser scanningunit for reducing the area of a spot projected onto the target toachieve the effect of improving the resolution.

A further object of the present invention is to provide a two-element fθlens used for a MEMS laser scanning unit, and the two-element fθ lenscan make a distortion correction to correct optical axis caused by thedeviation of the scan light resulting in the problems of an increaseddeviation of the main scanning direction and the sub scanning direction,and a change of a spot of a drum at the image into an oval like shape,and the two-element fθ lens can uniform the size of each image spot toachieve the effect of enhancing the image quality.

Therefore, the two-element lens used for a micro-electro mechanicalsystem (MEMS) laser scanning unit of the invention is applicable for alight source comprising an emitting laser beam, wherein a resonantoscillation is used for reflecting the laser beam of the light sourceonto MEMS reflecting mirror of the scan light to form an image on thetarget. As to a laser printer, the target is generally a drum. The spotof the image forms a scan light after the laser beam is emitted from thelight source, scanned oscillatory by the MEMS reflecting mirror, andreflected by the MEMS reflecting mirror. After the angle and position ofthe scan light are corrected by the two-element fθ lens of theinvention, a spot is formed on the drum. Since a photosensitive agent iscoated onto the drum, data can be printed out on a piece of paper by thesensing carbon powder centralized on the paper.

The two-element fθ lens of the invention comprises a first lens and asecond lens, counted from the MEMS reflecting mirror, wherein the firstlens includes a first optical surface and a second optical surface, thesecond lens includes a third optical surface and a fourth optical. Theseoptical surface provided the functions of correcting the phenomenon ofnon-uniform speed scanning which results in decreasing or increasing thedistance between spots on an image surface of a MEMS reflecting mirrorwith a simple harmonic movement with time into a constant speedscanning, so that the projection of a laser beam onto an image side cangive a constant speed scanning, and uniform the deviation of imageformed on the drum which caused by a scan light in the main scanningdirection and the sub scanning direction deviated from the optical axis,so as to make the correction to focus the scan light at a target.

To make it easier for our examiner to understand the technicalcharacteristics and effects of the present invention, we use preferredembodiments and related drawings for the detailed description of thepresent invention as follows.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows optical paths of a two-element fθ lens of the presentinvention;

FIG. 2 shows a relation of scanning angle θ versus time t of a MEMSreflecting mirror;

FIG. 3 shows an optical path chart and numerals of a scan light passingthrough a first lens and a second lens;

FIG. 4 shows a spot area varied with a different projecting positionafter a scan light is projected onto a drum;

FIG. 5 shows the Y direction of Gaussian beam diameter of scanning lightemitted by fθ lens;

FIG. 6 shows an optical path chart of a scan light passing through afirst lens and a second lens;

FIG. 7 shows spots in accordance with a first preferred embodiment ofthe present invention;

FIG. 8 shows spots in accordance with a second preferred embodiment ofthe present invention;

FIG. 9 shows spots in accordance with a third preferred embodiment ofthe present invention;

FIG. 10 shows spots in accordance with a fourth preferred embodiment ofthe present invention; and

FIG. 11 shows spots in accordance with a fifth preferred embodiment ofthe present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to FIG. 1 for a schematic view of optical paths of atwo-element fθ lens used for micro-electro mechanical system (MEMS)laser unit in accordance with the present invention, the two-element fθlens used for a micro-electro mechanical system (MEMS) laser scanningunit comprises: a first lens 131 having a first optical surface 131 aand a second optical surface 131 b, and a second lens having a thirdoptical surface 132 a and a fourth optical surface 132 b, which areapplicable for a MEMS laser scanning unit. In FIG. 1, the MEMS laserscanning unit comprises a laser source 11, a MEMS reflecting mirror 10,a cylindrical lens 16, two photoelectric sensors 14 a, 14 b and a lightsensing target. In FIG. 1 the target is achieved by a drum 15. After abeam 111 produced by the light laser source 11 is passed through acylindrical lens 16, the beam 111 is projected onto the MEMS reflectingmirror 10. The MEMS reflecting mirror 10 generates a resonantoscillation to reflect the beam 111 into scan lights 113 a, 113 b, 114a, 114 b, 115 a, 115 b at different time frames along the direction ofZ, wherein the scan lights 113 a, 113 b, 114 a, 114 b, 115 a, 115 b areprojected in a X direction which is called a sub scanning direction, andprojected in a Y direction which is called a main scanning direction,and the maximum scanning angle of the MEMS reflecting mirror 10 is θc.

Since the MEMS reflecting mirror 10 comes with a simple harmonicmovement, and the angle of movement shows a sinusoidal change with timeas shown in FIG. 2, therefore the angle and time of reflecting the scanlight are in a non-linear relation. The swinging angle of the MEMSreflecting mirror 10 has a wave peak a-a′ and a wave valley b-b′ asshown in the figure, and its swinging angle is significantly smallerthan the wave sections a-b an a′-b′, and this non-uniform angular speedmay cause an image deviation easily produced on the drum 15 by the scanlight. Therefore, a photoelectric sensor 14 a, 14 b are installed at theangle ±θp within range below the maximum scanning angle ±θc of the MEMSreflecting mirror 10 and the laser beam 111 starts to be reflected bythe MEMS reflecting mirror 10 at the wave peak as shown in FIG. 2, whichis equivalent to the scan light 115 a as shown in FIG. 1. If thephotoelectric sensor 14 a detects a scanned beam, it means that the MEMSreflecting mirror 10 swings to an angle of ±θp, which is equivalent tothe scan light 114 a as shown in FIG. 1. If the MEMS reflecting mirror10 scans point “a” at an angle variation as shown in FIG. 2, such pointis equivalent to the position if the scan light 113 a. Now, the lasersource 11 is controlled to start emitting the laser beam 111. When thepoint “b” as shown in FIG. 2 is scanned, such point is equivalent to theposition of the scan light 113 b (which is equivalent to the laser beam111 emitted by the laser source 11 a within an angle of ±θn). When theMEMS reflecting mirror 10 swings in an opposite direction to a wavesection a′-b′, the laser source 11 is controlled to start emitting thelaser beam 111 to complete a cycle.

Referring to FIG. 3 for an optical path chart of a scan light passingthrough a first lens and a second lens, ±θn is a valid scanning angle.If the MEMS reflecting mirror 10 is swinged to an angle of ±θn, thelaser source 11 starts emitting the desired scanning laser beam 111which is reflected into a scan light by the MEMS reflecting mirror 10,and the scan light is passed through the first lens 131 and refracted bythe first optical surface and the second optical surface of the firstlens 131, and the scan light reflected by the MEMS reflecting mirror 10with a none-linear relation of distance and time is converted into ascan light with a linear relation of distance and time. After the scanlight is passed through the first lens 131 and the second lens 132, thefocusing effect of the first optical surface 131 a, the second opticalsurface 131 b, the third optical surface 132 a and the fourth opticalsurface 132 b of the first lens 131 and the second lens 132 and theinterval of each optical surface can focus the scan light at the drum 15and form a column of spots 2 on the drum 15, and the distance betweenthe farthest two spots projected on the drum 15 is called an effectivescanning windows 3, wherein along the optical axis Z, d₁ is the distancebetween the MEMS reflecting mirror 10 and the first optical surface, d₂is the distance between the first optical surface and the second opticalsurface, d₃ is the distance between the second optical surface, R₂ andthe third optical surface, d₄ is the distance between the third opticalsurface and the fourth optical surface R₄, d₅ is the distance betweenthe fourth optical surface and the drum, R₁ is the radius of curvatureof the first optical surface, R₂ is the radius of curvature of thesecond optical surface, R₃ is the radius of curvature of the thirdoptical surface, R₄ is the radius of curvature of the fourth opticalsurface in the optical axis.

Referring to FIG. 4 for a spot area varied with a different projectingposition after a scan light is projected onto a drum, if the scan light113 a is projected in a different along the optical axis Z and onto thedrum 15 by the first lens 131 and the second lens 132, the incidentangles of the first lens 131 and the second lens 132 are zero, and thusthe deviation of the main scanning direction is minimum (said zero), andthe image at the spot 2 a on the drum 15 is in an inferenced circle-likeshape (same shape as laser light beam). After the scan light 113 b and113 c is projected on the drum 15 by the first lens 131 and the secondlens 132, the incident angle of the first lens 131 and the second lens132 with respect to the optical axis is non-zero, and the deviation ofthe main scanning direction is non-zero, and thus the projectiondistance of the main scanning direction is longer than the spot formedby the scan light 111 a is also bigger. Not only has the phenomenonexisted in the main scanning direction but also in the sub scanningdirection. Therefore, the image at the spot 2 b,2 c on the drum 15 is inan oval-like shape, and the area of 2 b, 2 c is greater than the area of2 a. Denoted S_(a0) and S_(b0) are the lengths of spots of the scanlight in the main scanning direction (Y direction) and the sub scanningdirection (X direction) on a reflecting surface of the MEMS reflectingmirror 10, and G_(a0) and G_(b0) are the Gaussian beam diameter ofscanning light emitted by fθ lens 13 at the intensity is 13.5% ofmaximum intensity on Y direction and the X direction, illustrated byFIG. 5. In FIG. 5, only Y direction Gaussian beam is shown. Thetwo-element fθ lens of the invention can control the spot size in themain scanning direction within a limited range by the distortioncorrection of the fθ lens 13 and correct the spot size in the subscanning direction by the distortion correction of the first lens 131and the second 132 of the two-element fθ lens 13, such that the spotsize is controlled within a limited range, and the distribution of thespot size (or the ratio of largest spots and smallest spots) iscontrolled within an appropriate range in compliance with the requiredresolution.

To achieve the forgoing effects, the two-element fθ lens of theinvention comes with a first lens having a first optical surface and asecond optical lens having a third optical surface and a fourth opticalsurface of with a spherical surface or an aspherical surface. If theAspherical surface is adopted, the aspherical surface is designed withthe following equations (2) or (3)

1. Anamorphic Equation

$\begin{matrix}{Z = {\frac{{({Cx})X^{2}} + {({Cy})Y^{2}}}{1 + \sqrt{1 - {\left( {1 + {Kx}} \right)({Cx})^{2}X^{2}} - {\left( {1 + {Ky}} \right)({Cy})^{2}Y^{2}}}} + {A_{R}\left\lbrack {{\left( {1 - A_{P}} \right)X^{2}} + {\left( {1 + A_{P}} \right)Y^{2}}} \right\rbrack}^{2} + {B_{R}\left\lbrack {{\left( {1 - B_{P}} \right)X^{2}} + {\left( {1 + B_{P}} \right)Y^{2}}} \right\rbrack}^{3} + {C_{R}\left\lbrack {{\left( {1 - C_{P}} \right)X^{2}} + {\left( {1 + C_{P}} \right)Y^{2}}} \right\rbrack}^{4} + {D_{R}\left\lbrack {{\left( {1 - D_{P}} \right)X^{2}} + {\left( {1 + D_{P}} \right)Y^{2}}} \right\rbrack}^{5}}} & (2)\end{matrix}$

where, Z is the sag of any point on the surface parallel to the Z-axis,C_(x) and C_(y) are curvatures in the X direction and the Y directionrespectively, K_(x), and K_(y) are the conic coefficients in the Xdirection and the Y direction respectively and correspond to eccentriccity in the same way as conic coefficient for the Aspherical surfacetype, A_(R), B_(R), C_(R) and D_(R) are deformations from the coniccoefficient of rotationally symmetric portions of the fourth order, thesixth order, the eighth and the tenth order respectively, and A_(P),B_(P), C_(P) and D_(P) are deformation from the conic coefficient ofnon-rotationally symmetric components to the fourth order, the sixthorder, the eight order and the tenth order respectively. This reduces toAspherical surface type when C_(x)=C_(y), K_(x)=K_(y) andA_(P)=B_(p)=C_(p)=D_(p)=0

2. Toric Equation

$\begin{matrix}\begin{matrix}{Z = {{Zy} + \frac{({Cxy})X^{2}}{1 + \sqrt{1 - {({Cxy})^{2}X^{2}}}}}} \\{{Cxy} = \frac{1}{\left( {1/{Cx}} \right) - {Zy}}} \\{{Zy} = {\frac{({Cy})Y^{2}}{1 + \sqrt{1 - {\left( {1 + {Ky}} \right)({Cy})^{2}Y^{2}}}} + {B_{4}Y^{4}} + {B_{6}Y^{6}} + {B_{8}Y^{8}} + {B_{10}Y^{10}}}}\end{matrix} & (3)\end{matrix}$

where, Z is the zag pf any point on the surface parallel to the Z-axis;C_(y) and C_(x) are curvatures in the X direction and the Y directionrespectively, K_(y) is a conic coefficient in the Y direction, B₄, B₆,B₈ and B₁₀ are deformations from the conic coefficient to the fourth,sixth, eight and tenth order respectively. When C_(x)=C_(y) andK_(y)=A_(P)=B_(p)=C_(p)=D_(p)=0 is reduced to a single sphericalsurface.

To uniform the scan speed of the scan light projected onto the image ofthe target, the invention adopts two equal time interval and an equaldistance between two spots, and the two-element fθ lens of the inventioncan correct the emergence angle of the scan light between the scan light113 a to the scan light 113 b, so that the first lens 131 and the secondlens 132 corrects the emergence angle of the scan light to produce twoscan lights at the same time interval. After the emergence angle iscorrected, the distance between any two spots formed on the drum 15 ofthe image is equal. Further, after the laser beam 111 is reflected bythe MEMS reflected mirror 10, the spot is diverged and becomes larger.After the scan light is passed through the distance from the MEMSreflecting mirror 10 to the drum 15, the spot becomes larger. Sucharrangement is incompliance with the actual required resolution. Thetwo-element fθ lens of the invention further focuses from the scan light113 a to the scan light 113 b reflected by the MEMS reflecting mirror 10at the drum 15 of the image to from a smaller spot in the main scanningand sub scanning directions. The two-element fθ lens of the inventionfurther uniforms the spot size of the image on the drum 15 (to limitspot size in a range to comply with the required resolution) for thebest condition.

The two-element fθ lens comprises a first lens 131 and a second lens132, counted from the MEMS reflecting mirror 10, and the first lens is apositive power meniscus lens of which the concave surface is disposed ona side of a MEMS mirror, the second lens is a negative power meniscuslens of which the convex surface is disposed on the side of the MEMSmirror, wherein the first lens 131 includes a first optical surface 131a and a second optical surface 131 b for converting a scan spot with anon-linear relation of angle with time and reflected by the MEMSreflecting mirror 10 into a scan spot with a linear relation of distancewith time; and the second lens 132 includes a third optical surface 132a and a fourth optical surface 132 b for correcting the focus of thescan light of the first lens 131 onto target; such that the two-elementfθ lens projects a scan light reflected by the MEMS reflecting mirror 10onto the image of the drum 15. The first optical surface 131 a, thesecond optical surface 131 b, the third optical surface 132 a and thefourth optical surface 132 b are optical surfaces are composed of atleast one Aspherical surface in the main scanning direction. The firstoptical surface 131 a and the second optical surface 131 b are opticalsurfaces composed of at least one aspherical surface in the sub scanningdirection. Further, the assembly of the first lens 131 and the secondlens 132 of the two-element fθ lens in accordance with the presentinvention has an optical effect in the main scanning direction thatsatisfies the conditions of Equation (4) and (5):

$\begin{matrix}{0.1 < \frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}} < 1.2} & (4)\end{matrix}$

$\begin{matrix}{{- 0.6} < \frac{d_{5}}{f_{{(2)}Y}} < {- 0.01}} & (5)\end{matrix}$

or, the main scanning direction satisfies the conditions of equation (6)

$\begin{matrix}{0.3 < {{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right)}} < 0.6} & (6)\end{matrix}$and the sub scanning direction satisfies the conditions of equation (7)

$\begin{matrix}{0.1 < {{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}} < 1.1} & (7)\end{matrix}$

where, f_((1)Y) is the focal length of the first lens 131 in the mainscanning direction, f_((2)Y) is the focal length of the second lens 132in the main scanning direction, d₃ is the distance between an opticalsurface on a target side of the first lens 131 when θ=0° and an opticalsurface on the MEMS reflecting mirror side of the second lens 132, d₄ isthe thickness of the second lens when θ=0°, d₅ is the distance betweenan optical surface on a target side of the second lens 132 when θ=0° andthe target, f_((1)X) is the focal length of the first lens in the subscanning direction, f_((2)X) is the focal length of the second lens inthe sub scanning direction, f_(s) is the combined focal length of thetwo-element fθ lens, R_(ix) is the radius of curvature of the i-thoptical surface in the X direction; and nd1 and n_(d2) are therefraction indexes of the first lens and the second lens 13respectively.

Further, the spot uniformity produced by the two-element fθ lens of theinvention can be indicated by the ratio δ of the largest spot and thesmallest spot size that satisfies the conditions of Equation (8):

$\begin{matrix}{{0.8 < \delta} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}} & (8)\end{matrix}$

The resolution produced by the two-element fθ lens of the invention canbe indicated by the ratio η_(max) of the largest spot on the drum 15formed by the scan light on the reflecting surface of the MEMSreflecting mirror 10 (or the ratio of scanning light of maximum spot)and the ratio η_(min) of the smallest spot formed by the scan light onthe reflecting surface of the MEMS reflecting mirror 10 (or the ratio ofscanning light of minimum spot), and the ratios satisfy the conditionsof Equations (9) and (10)

$\begin{matrix}{\eta_{\max} = {\frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)} < 0.10}} & (9)\end{matrix}$

$\begin{matrix}{\eta_{\min} = {\frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)} < 0.10}} & (10)\end{matrix}$

where, S_(a) and S_(b) are the lengths of any one spot of the scan lightformed on the drum in the main scanning direction and the sub scanningdirection, δ is the ratio of the smallest spot and the largest spot onthe drum 15, S_(a0) and S_(b0) are the lengths of the spots of the scanlight on the reflecting surface of the MEMS reflecting mirror 10 in themain scanning direction and the sub scanning direction respectively.

To make it easier for our examiner to understand the structure andtechnical characteristics of the present invention, we use the preferredembodiments accompanied with related drawings for the detaileddescription of the present invention as follows.

The following preferred embodiments of the invention disclose atwo-element fθ lens used for a micro-electro mechanical system (MEMS)laser scanning unit by using major elements for the illustration, andthus the preferred embodiments can be applied in a MEMS laser scanningunit including but not limited to the two-element fθ lens withcomponents illustrated in the embodiments only, but any otherequivalents are intended to be covered in the scope of the presentinvention. In other words, any variation and modification of thetwo-element fθ lens used for a micro-electro mechanical system (MEMS)laser scanning unit can be made by the persons skilled in the art. Forexample, the radius of curvature of the first lens and the second lens,the design of the shape, the selected material and the distance can beadjusted without any particular limitation.

In a first best embodiment, a first lens and a second lens of thetwo-element fθ lens are lens, the first lens is a positive powermeniscus lens of which the concave surface is disposed on a side of aMEMS mirror, the second lens is a negative power meniscus lens of whichthe convex surface is disposed on the side of the MEMS mirror, and afirst optical surface of the first lens is a Spherical surfaces, and asecond optical surface of the first lens, a third optical surface and afourth optical surface of the second lens are Aspherical surfacesdesigned in accordance with the Equation (2), and the opticalcharacteristics and the Aspherical surface parameters are listed inTables 1 and 2.

TABLE 1 Optical Characteristics of fθ lens for First PreferredEmbodiment optical surface radius (mm) d, thickness (mm) nd, refractionindex MEMS surface ∞ 11.65 1 R0 lens 1 1.527 R1 R1x 143.33 13.04 R1y−62.25 R2 (Anamorphic) R2x* −15.35 22.00 R2y* −36.88 lens 2 1.527 R3(Anamorphic) R3x* 19.89 12.18 R3y* 223.38 R4 (Anamorphic) R4x* 75.5289.76 R4y* 101.98 drum surface R5 ∞ 0.00 *denoted aspherical surface

TABLE 2 Parameters of Aspherical Surface of Optical Surface Parameterfor First Preferred Embodiment Anamorphic equation coefficient 4th Order6th Order 8th Order 10th Order optical Ky Conic Coefficient CoefficientCoefficient Coefficient surface Coefficient (AR) (BR) (CR) (DR) R2*−1.0639E+00 −2.4178E−06   0.0000E+00 0.0000E+00 0.0000E+00 R3*−4.9963E+01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 R4* −6.1801E+000.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 4th Order 10th Order opticalKx Conic Coefficient 6th Order 8th Order Coefficient surface Coefficient(AP) Coefficient (BP) Coefficient (CP) (DP) R2* −5.3942E−01 5.1047E−020.0000E+00 0.0000E+00 0.0000E+00 R3* −2.9071E+00 0.0000E+00 0.0000E+000.0000E+00 0.0000E+00 R4*   1.1328E+01 0.0000E+00 0.0000E+00 0.0000E+000.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=145.78, f_((2)Y)=−368.67,f_(sX)=23.655, f_(sY)=215.37 (mm), so that the scan light can beconverted into a scan spot with a linear relation of distance and time,and the spots with spot 3 S_(a0)=13.642 and S_(b0)=3718.32 (μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 3. The maximum diameter (μm) ofgeometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 4. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 7, where the diameter of unity circle is 0.05 mm.

TABLE 3 Conditions for first Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.8502$\frac{d_{5}}{f_{{(2)}Y}}$ −0.2435${Main}\mspace{14mu}{scanning}{{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.4707${Sub}\mspace{14mu}{scanning}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}}$0.9481$\delta = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}$0.8787$\eta_{\max} = \frac{\max\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b0} \cdot S_{a0}} \right)}$0.0685$\eta_{\min} = \frac{\min\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b0} \cdot S_{a0}} \right)}$0.0602

TABLE 4 The maximum diameter (μm) of light spot on the drum Y −107.460−96.206 −84.420 −96.206 −60.206 −48.050 −35.947 −23.914 0.000 Maxdiameter 4.70E−03 3.75E−03 3.33E−03 3.48E−03 3.96E−03 4.13E−03 4.02E−033.43E−03 2.77E−03

In a second best embodiment, a first lens and a second lens of thetwo-element fθ lens are lens, the first lens is a positive powermeniscus lens of which the concave surface is disposed on a side of aMEMS mirror, the second lens is a negative power meniscus lens of whichthe convex surface is disposed on the side of the MEMS mirror, and afirst optical surface and a second optical surface of the first lens areSpherical surfaces, a third optical surface of the second lens is anAspherical surface designed in accordance with the Equation (2), and afourth optical surface of the second lens is an Aspherical surfacedesigned in accordance with the Equation (3), and the opticalcharacteristics and the Aspherical surface parameters are listed inTables 5 and 6.

TABLE 5 Optical characteristics of fθ lens for Second PreferredEmbodiment. d, optical surface radius (mm) thickness (mm) nd, refractionindex MEMS surface R0 ∞ 12.42 1 lens 1 1.527 R1 R1x 107.63 12.59 R1y−51.38 R2 R2x −15.74 11.37 R2y −32.25 lens 2 1.527 R3 (Anamorphic) R3x*19.26 8.00 R3y* 75.91 R4 (Y Toroid) R4x 70.85 99.56 R4y* 45.26 drumsurface R5 ∞ 0.00 *denoted aspherical surface

TABLE 6 Parameters of Aspherical Surface of Optical Surface Parameterfor Second Preferred Embodiment Toric equation Coefficient 10th Orderoptical Ky (Conic 4th Order 6th Order 8th Order Coefficient surfaceCoefficient) Coefficient (B4) Coefficient (B6) Coefficient (B8) (B10)R4* −7.2939E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 Anamorphicequation Coefficient 4th Order 6th Order 8th Order 10th Order optical Ky(Conic Coefficient Coefficient Coefficient Coefficient surfaceCoefficient) (AR) (BR) (CR) (DR) R3* −2.2659E+01 0.0000E+00 0.0000E+000.0000E+00 0.0000E+00 4th Order 10th Order Kx (Conic Coefficient 6thOrder 8th Order Coefficient Coefficient) (AP) Coefficient (BP)Coefficient (CP) (DP) R3*   2.1601E−01 0.0000E+00 0.0000E+00 0.0000E+000.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=133.89, f_((2)Y)=−233.70,f_(sX)=20.84, f_(sY)=274.205 (mm), so that the scan light can beconverted into a scan sopt with a linear relation of distance and time,and the spots with spot 3 S_(a0)=13.824 and S_(b0)=3512.066 (μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 7. The maximum diameter (μm) ofgeometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 8. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 8, where the diameter of unity circle is 0.05 mm.

TABLE 7 Conditions for Second Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.8883$\frac{d_{5}}{f_{{(2)}Y}}$ −0.4260${Main}{\mspace{11mu}\;}{scanning}{{f_{s\; Y} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.4609${Sub}\mspace{14mu}{scanning}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{s\; X}}}}$0.8321$\delta = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}$0.8802$\eta_{\max} = \frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0894$\eta_{\min} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0787

TABLE 8 The maximum diameter (μm) of geometric spot on the drum Y−107.460 −96.206 −84.420 −96.206 −60.206 −48.050 −35.947 −23.914 0.000Max diameter 1.35E−02 1.27E−02 1.21E−02 1.28E−02 1.35E−02 1.41E−021.42E−02 1.37E−02 1.22E−02

In a third best embodiment, a first lens and a second lens of thetwo-element fθ lens are lens, the first lens is a positive powermeniscus lens of which the concave surface is disposed on a side of aMEMS mirror, the second lens is a negative power meniscus lens of whichthe convex surface is disposed on the side of the MEMS mirror, and afirst optical surface of the first lens is a Spherical surface, a secondoptical surface, a third optical surface and a fourth optical surface ofthe second lens are Aspherical surfaces designed in accordance with theEquation (2), and the optical characteristics and the Aspherical surfaceparameters are listed in Tables 9 and 10.

TABLE 9 Optical Characteristics of fθ Lens for Third PreferredEmbodiment d, optical surface radius (mm) thickness (mm) nd, refractionindex MEMS surface R0 ∞ 19.84 1 lens 1 1.527 R1 R1x −388.85 11.22 R1y−112.39 R2 (Anamorphic) R2x* −15.41 15.00 R2y* −42.77 lens 2 1.527 R3(Anamorphic) R3x* 25.94 12.00 R3y* 422.59 R4 (Anamorphic) R4x* 56.9394.18 R4y* 125.67 drum surface R5 ∞ 0.00 *denoted aspherical surface

TABLE 10 Parameters of Aspherical Surface for Third Preferred EmbodimentAnamorphic equation Coefficient 4th Order 6th Order 8th Order 10th Orderoptical Ky Conic Coefficient Coefficient Coefficient Coefficient surfaceCoefficient (AR) (BR) (CR) (DR) R2* −1.1079E+00 −1.8898E−06   0.0000E+000.0000E+00 0.0000E+00 R3* −2.0505E+02 0.0000E+00 0.0000E+00 0.0000E+000.0000E+00 R4* −4.9535E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+004th Order 10th Order Kx Conic Coefficient 6th Order 8th OrderCoefficient Coefficient (AP) Coefficient (BP) Coefficient (CP) (DP) R2*−4.9292E−01 −5.6828E−02   0.0000E+00 0.0000E+00 0.0000E+00 R3*−4.9518E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 R4*   2.1264E+000.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=124.07, f_((2)Y)=−344.01,f_(sX)=23.785, f_(sY)=176.355 (mm), so that the scan light can beconverted into a scan spot with a linear relation of distance and time,and the spots with spot 3 S_(a0)=13.452 and S_(b0)=3941.106 (μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 11. The maximum diameter (μm)of geometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 12. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 9, where the diameter of unity circle is 0.05 mm.

TABLE 11 Conditions for Third Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.9768$\frac{d_{5}}{f_{{(2)}Y}}$ −0.2738${Main}\mspace{14mu}{scanning}\mspace{14mu}{{f_{s\; Y} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.4789${Sub}\mspace{14mu}{scanning}\mspace{14mu}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{s\; X}}}}$0.5615$\delta = \frac{\min\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}{{\max\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}\mspace{14mu}}$0.8776$\eta_{\max} = \frac{\max\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b0} \cdot S_{a0}} \right)}$0.0586$\eta_{\min} = \frac{\min\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b0} \cdot S_{a\; 0}} \right)}$0.0515

TABLE 12 The maximum diameter (μm) of geometric spot on the drum Y−107.458 −96.173 −84.419 −96.173 −60.343 −48.232 −36.136 −24.067 0.000Max diameter 3.75E−03 2.27E−03 1.89E−03 1.96E−03 3.05E−03 3.73E−033.92E−03 3.40E−03 1.84E−03

In a fourth best embodiment, a first lens and a second lens of thetwo-element fθ lens are lens, the first lens is a positive powermeniscus lens of which the concave surface is disposed on a side of aMEMS mirror, the second lens is a negative power meniscus lens of whichthe convex surface is disposed on the side of the MEMS mirror, and afirst optical surface is a Spherical surface, a second optical surfaceof the first lens and a third optical surface of the second lens areAspherical surfaces designed in accordance with the Equation (2), afourth optical surface of the second lens is an Aspherical surfacedesigned in accordance with the Equation (3), and the opticalcharacteristics and the Aspherical surface parameters are listed inTables 13 and 14.

TABLE 13 Optical Characteristics of fθ lens for Fourth PreferredEmbodiment d, optical surface radius (mm) thickness (mm) nd, refractionindex MEMS surface R0 ∞ 12.49 1 lens 1 1.527 R1 R1x 79.81 11.98 R1y−48.62 R2 (Anamorphic) R2x −15.47 10.00 R2y* −31.46 lens 2 1.527 R3(Anamorphic) R3x 19.60 8.00 R3y* 62.12 R4 (Y Toroid) R4x 71.71 101.12R4y* 40.00 drum surface R5 ∞ 0.00 *denoted aspherical surface

TABLE 14 Parameters of Aspherical Surface of Optical Surface for FourthPreferred Embodiment Toric equation Coefficient 10th Order optical Ky(Conic 4th Order 6th Order 8th Order Coefficient surface Coefficient)Coefficient (B4) Coefficient (B6) Coefficient (B8) (B10) R4* −6.7983E+000.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 Anamorphic equationCoefficient 4th Order 6th Order 8th Order 10th Order optical Ky (ConicCoefficient Coefficient Coefficient Coefficient surface Coefficient)(AR) (BR) (CR) (DR) R2* −5.7584E−01 0.0000E+00 0.0000E+00 0.0000E+000.0000E+00 R3* −1.7523E+01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=136.21, f_((2)Y)=−243.44,f_(sX)=19.258, f_(sY)=270.784 (mm), so that the scan light can beconverted into a scan spot with a linear relation of distance and time,and the spots with spot 3 S_(a0)=13.81 and S_(b0)=35220.04 (μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 15. The maximum diameter (μm)of geometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 16. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 10, where the diameter of unity circle is 0.05 mm.

TABLE 15 Conditions for Fourth Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.8745$\frac{d_{5}}{f_{{(2)}Y}}$ −0.4157${Main}\mspace{14mu}{scanning}\mspace{14mu}{{f_{s\; Y} \cdot \left( {\frac{\left( {n_{d1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.4615${Sub}\mspace{14mu}{scanning}{\mspace{14mu}{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{s\; X}}}}$0.7910$\delta = \frac{\min\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}{\max\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}$0.8772$\eta_{\max} = \frac{\max\mspace{14mu}\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0945$\eta_{\min} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0829

TABLE 16 The maximum diameter (μm) of geometric spot on target drum Y−107.460 −96.206 −84.420 −96.206 −60.206 −48.050 −35.947 −23.914 0.000Max diameter 1.38E−02 1.28E−02 1.21E−02 1.28E−02 1.35E−02 1.40E−021.43E−02 1.39E−02 1.26E−02

In a fifth best embodiment, a first lens and a second lens of thetwo-element fθ lens are lens, the first lens is a positive powermeniscus lens of which the concave surface is disposed on a side of aMEMS mirror, the second lens is a negative power meniscus lens of whichthe convex surface is disposed on the side of the MEMS mirror, and asecond optical surface of the first lens and a third optical surface ofthe second lens are Aspherical surfaces designed in accordance with theEquation (2), a first optical surface of the first lens and a fourthoptical surface of the second lens are Aspherical surfaces designed inaccordance with the Equation (3), and the optical characteristics andthe Aspherical surface parameters are listed in Tables 17 and 18.

TABLE 17 Optical Characteristics of fθ Lens for Fourth PreferredEmbodiment d, optical surface radius (mm) thickness (mm) nd, refractionindex MEMS surface R0 ∞ 30.74 1 lens 1 1.527 R1 (Y Toroid) R1x −41.7310.00 R1y* −39.34 R2 (Anamorphic) R2x* −11.19 12.95 R2y* −39.34 lens 21.527 R3 (Anamorphic) R3x* 347.39 12.00 R3y* 140.92 R4 (Y Toroid) R4x99.54 76.38 R4y* 124.52 drum surface R5 ∞ 0.00 *denoted asphericalsurface

TABLE 18 Parameters of Aspherical Surface of Optical Surface for FifthPreferred Embodiment Toric equation Coefficient 10th Order optical Ky(Conic 4th Order 6th Order 8th Order Coefficient surface Coefficient)Coefficient (B4) Coefficient (B6) Coefficient (B8) (B10) R1*  2.3701E−01 −1.1545E−07 −5.1670E−10   0.0000E+00 0.0000E+00 R4*−1.5112E+01 −5.2786E−09 4.4640E−12 0.0000E+00 0.0000E+00 Anamorphicequation Coefficient 6th Order 8th Order 10th Order optical Ky (Conic4th Order Coefficient Coefficient Coefficient surface Coefficient)Coefficient (AR) (BR) (CR) (DR) R2* −1.1425E−01 −2.2983E−07−2.1475E−10   0.0000E+00 0.0000E+00 R3* −2.1300E+01 −1.2206E−065.2723E−12 0.0000E+00 0.0000E+00 8th Order 10th Order Kx Conic 4th Order6th Order Coefficient Coefficient Coefficient Coefficient (AP)Coefficient (BP) (CP) (DP) R2* −7.6546E−01 −1.0556E+00 0.0000E+000.0000E+00 0.0000E+00 R3*   1.0000E+01 −8.3557E−01 0.0000E+00 0.0000E+000.0000E+00

Referring to FIG. 6 for the optical path chart of an optical surface ofthe two-element fθ lens 13, f_((1)Y)=851.41, f_((2)Y)=−2714.78,f_(sX)=26.469, f_(sY)=1221.728 (mm), so that the scan light can beconverted into a scan spot with a linear relation of distance and time,and the spots with spot 3 S_(a0)=14.31 and S_(b0)=2983.85 (μm) on theMEMS reflecting mirror 10 are scanned into scan lights and focused onthe drum 15 to form a smaller spot 6 and satisfy the conditions ofEquations (4) to (10) as listed in Table 19. The maximum diameter (μm)of geometric spot on the drum at distance Y (mm) from the center pointalong the drum surface is shown in Table 20. The distribution of spotsizes from the central axis to the left side of the scan window 3 isoutlined as FIG. 11, where the diameter of unity circle is 0.05 mm.

TABLE 19 Conditions for Fifth Preferred Embodiment$\frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}}$ 0.1190$\frac{d_{5}}{f_{{(2)}Y}}$ −0.0281${Main}\mspace{14mu}{Scanning}{{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right.}}$0.4615${Sub}\mspace{14mu}{Scanning}{{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{s\; X}}}}$0.1243$\delta = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}$0.8435$\eta_{\max} = \frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0843$\eta_{\min} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)}$0.0711

TABLE 20 The maximum diameter (μm) of geometric spot on target drum Y−107.460 −96.206 −84.420 −96.206 −60.206 −48.050 −35.947 −23.914 0.000Max diameter 1.35E−02 1.27E−02 1.21E−02 1.28E−02 1.35E−02 1.41E−021.42E−02 1.37E−02 1.22E−02

In view of the aforementioned preferred embodiments, the presentinvention at least has the following effects:

(1) With the two-element fθ lens of the invention, the scanning iscorrected the phenomenon of non-uniform speed which results indecreasing or increasing the distance between spots on an image surfaceof a MEMS reflecting mirror with a simple harmonic movement with timeinto a constant speed scanning, so that the laser beam at the image sideis projected for a uniform speed scanning and an equal distance betweenany two adjacent spots can be achieved for the image on a target.(2) With the two-element fθ lens of the invention, the distortioncorrection is provided for correcting the main scanning direction andsub scanning direction of the scan light, so that the image size of thespot focused at the target can be decreased.(3) With the two-element fθ lens of the invention, the distortioncorrection is provided for correcting the main scanning direction andthe sub scanning direction of the scan light, so as to focus the spotsize focused and imaged at the target.

1. A two-element fθ lens used for a micro-electro mechanical system(MEMS) laser scanning unit, said MEMS laser scanning unit comprising alight source for emitting laser beam, a MEMS reflecting mirror forreflecting said laser beam emitted by said light source into a scanninglight by resonant oscillation, and a target provided for sensing light,said two-element fθ lens being disposed between said target and saidMEMS reflecting mirror, said two-element fθ lens comprising: a firstlens, in a positive power meniscus shape, and having a concave surfacetoward said MEMS reflecting mirror; and a second lens, in a negativepower meniscus shape, and having a convex surface toward said MEMSreflecting mirror, located between said first lens and said target;wherein, said first lens included a first optical surface and a secondoptical surface, at least one of said optical surfaces is an asphericalsurface in both main scanning direction and sub scanning direction ofsaid MEMS laser scanning unit; wherein, said second lens included athird optical surface and a fourth optical surface, at least one of saidoptical surfaces is an aspherical surface in both main scanningdirection and sub scanning direction of said MEMS laser scanning unit;wherein, said two-element fθ lens converts the non-linear relation ofreflecting angle with time of said scanning light into a linear relationbetween the distance of the scan spot with time and focusing thescanning light to form an image at said target.
 2. The two-element fθlens of claim 1, wherein the main scanning direction satisfies theconditions of:${{{{0.1 < \frac{d_{3} + d_{4} + d_{5}}{f_{{(1)}Y}} < 1.2};} - 0.6} < \frac{d_{5}}{f_{{(2)}Y}} < {- 0.01}};$wherein, f_((1)Y) is the focal length of the first lens in the mainscanning direction, and f_((2)Y) is the focal length of the second lensin the main scanning direction, and d₃ is the distance from the secondoptical surface to the third optical surface on the optical axis Z, andd₄ is the thickness of the second lens along the optical axis Z, and d₅is the distance from the fourth optical surface to the target side onthe optical axis Z.
 3. The two-element fθ lens of claim 1, furthersatisfying the conditions of: in the main scanning direction$0.3 < {{f_{sY} \cdot \left( {\frac{\left( {n_{d\; 1} - 1} \right)}{f_{{(1)}y}} + \frac{\left( {n_{d\; 2} - 1} \right)}{f_{{(2)}y}}} \right)}} < 0.6$and in the sub scanning direction$0.1 < {{\left( {\frac{1}{R_{1x}} - \frac{1}{R_{2x}}} \right) + {\left( {\frac{1}{R_{3x}} - \frac{1}{R_{4x}}} \right)f_{sX}}}} < 1.1$wherein, f_((1)Y) and f_((1)X) are the focal lengths of the first lensin the main scanning direction and the sub scanning directionrespectively, and f_((2)Y) and f_((2)X) are the focal lengths of thesecond lens in the main scanning direction and the sub scanningdirection respectively, f_(s) is a combined focal length of thetwo-element fθ lens, and R_(ix) is the radius of curvature of the i-thoptical surface in the X direction; and n_(d1) and n_(d2) are refractionindexes of the first lens and the second lens respectively.
 4. Thetwo-element fθ lens of claim 1, wherein the ratio of the largest spotand the smallest spot size satisfies the conditions of:${{0.8 < \delta} = \frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\max\left( {S_{b} \cdot S_{a}} \right)}};$wherein, S_(a) and S_(b) are the lengths of any spot formed by a scanlight on the target in the main scanning direction and the sub scanningdirection, and δ is the ratio of the smallest spot and the largest spoton the target.
 5. The two-element fθ lens of claim 1, wherein the ratioof the largest spot on the target and the smallest spot on the targetsatisfies the conditions of:${\eta_{\max} = {\frac{\max\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)} < 0.10}};$${\eta_{\min} = {\frac{\min\left( {S_{b} \cdot S_{a}} \right)}{\left( {S_{b\; 0} \cdot S_{a\; 0}} \right)} < 0.10}};$wherein, S_(a0) and S_(b0) are the lengths of a spot formed by a scanlight on a reflecting surface of the MEMS reflecting mirror in the mainscanning direction and the sub scanning direction, and Sa and S_(b) arethe lengths of any spot formed by a scan light on the target in the mainscanning direction and the sub scanning direction, and η_(max) is themaximum ratio value of the largest spot on the target with the spot onthe reflecting surface of the MEMS reflecting mirror, and η_(min) is theminimum ratio value of the largest spot on the target with the spot onthe reflecting surface of the MEMS reflecting mirror.